Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling
arXiv:1707.06489 · doi:10.1016/j.spa.2019.08.006
Abstract
A piecewise-deterministic Markov process, specified by random jumps and switching semi-flows, as well as the associated Markov chain given by its post-jump locations, are investigated in this paper. The existence of an exponentially attracting invariant measure and the strong law of large numbers are proven for the chain. Further, a one-to-one correspondence between invariant measures for the chain and invariant measures for the continuous-time process is established. This result, together with the aforementioned ergodic properties of the discrete-time model, is used to derive the strong law of large numbers for the process. The studied random dynamical systems are inspired by certain biological models of gene expression, which are also discussed within this paper.
38 pages, 1 figure
References in corpus (4)
Cited by in corpus (9)
- A Useful Version of the Central Limit Theorem for a General Class of Markov Chains
- The Strassen Invariance Principle for Certain Non-stationary Markov-Feller Chains
- The e-property of asymptotically stable Markov-Feller operators
- Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
- The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations
- Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process
- The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm
- On the Existence and Uniqueness of Stationary Distributions for Some Piecewise Deterministic Markov Processes With State-Dependent Jump Intensity
- Law of the Iterated Logarithm for Markov Semigroups with Exponential Mixing in the Wasserstein Distance